Introduction to Representation Theory, 1/3

• Birgit Speh (Cornell University)

Room: Amphitheater Darboux The 3 lectures are introduction to representation theory of reductive real Lie groups at the level of the book ”Representation theory of semisimple Lie groups: An overview based on examples” by A.W. Knapp. All the concepts and theorems will be illustrated by examples, but I will give references for the detailed proofs. In each lecture I will give some suggested homework and reading. Lecture 1. We will introduce discuss the basic important concepts: Admissible representations, differentiable vectors, equivalence of representations, irreducible representations, unitary representations, (g, K)-modules. Time permitting we will also introduce Cartan subgroups, the Iwasawa decomposition and parabolic subgroups.

Branching Problems in Representation Theory, 1/3

• Toshiyuki Kobayashi (The University of Tokyo)

Room: Amphitheater Darboux The concept of symmetries naturally arises in various areas of mathematics and science, including geometry, number theory, differential equations, and quantum mechanics. The more symmetries an object possesses, the better we can understand it through group-theoretic approaches. Branching problems investigate how large symmetries break down into smaller ones, such as fusion rules, using mathematical formulations based on the language of representations and their restrictions. These problems have been studied for over 80 years. In recent years, there has been a surge of research focused on the restriction of continuous symmetries in infinite-dimensional cases, leading to the development of new geometric and analytic methods. In my three lectures, I plan to provide an introduction to the branching problems of infinite-dimensional representations of real reductive groups, such as GL(n, R), using plenty of elementary examples to make the basic concepts and key ideas more accessible. A major goal of my lectures is to capitalize on this momentum and bring together young students and postdocs from diverse mathematical disciplines, encouraging them to develop an interest in this promising field. If time permits, I will also present some open questions in the area.

Different Aspects of Rankin–Cohen Operators, 1/2 (online)

• Michael Pevzner

Room: Amphitheater Darboux Rankin-Cohen brackets provide a fundamental example of “non-elementary” differential symmetry breaking operators. We will explore the combinatorial structure of these operators through multiple perspectives and discuss several approaches to find their higher dimensions analogues within the representation-theoretical perspective of branching problems.

Introduction to Representation Theory, 2/3

• Birgit Speh (Cornell University)

Room: Amphitheater Darboux The 3 lectures are introduction to representation theory of reductive real Lie groups at the level of the book ”Representation theory of semisimple Lie groups: An overview based on examples” by A.W. Knapp. All the concepts and theorems will be illustrated by examples, but I will give references for the detailed proofs. In each lecture I will give some suggested homework and reading. Lecture 2. We will discuss the representations of SU(2), SL(2, R) and of SL(2, C).

Different Aspects of Rankin–Cohen Operators, 2/2 (online)

• Michael Pevzner

Room: Amphitheater Darboux Rankin-Cohen brackets provide a fundamental example of “non-elementary” differential symmetry breaking operators. We will explore the combinatorial structure of these operators through multiple perspectives and discuss several approaches to find their higher dimensions analogues within the representation-theoretical perspective of branching problems.

Branching Problems in Representation Theory, 2/3

• Toshiyuki Kobayashi (The University of Tokyo)

Room: Amphitheater Darboux The concept of symmetries naturally arises in various areas of mathematics and science, including geometry, number theory, differential equations, and quantum mechanics. The more symmetries an object possesses, the better we can understand it through group-theoretic approaches. Branching problems investigate how large symmetries break down into smaller ones, such as fusion rules, using mathematical formulations based on the language of representations and their restrictions. These problems have been studied for over 80 years. In recent years, there has been a surge of research focused on the restriction of continuous symmetries in infinite-dimensional cases, leading to the development of new geometric and analytic methods. In my three lectures, I plan to provide an introduction to the branching problems of infinite-dimensional representations of real reductive groups, such as GL(n, R), using plenty of elementary examples to make the basic concepts and key ideas more accessible. A major goal of my lectures is to capitalize on this momentum and bring together young students and postdocs from diverse mathematical disciplines, encouraging them to develop an interest in this promising field. If time permits, I will also present some open questions in the area.

Differential Symmetry Breaking Operators and F-method

• Toshihisa Kubo (Ryukoku University)

Room: Amphitheater Darboux Let X be a smooth manifold and Y a smooth submanifold of X. Take G'⊂G to be a pair of Lie groups that act on Y⊂X, respectively. We call a differential operator D between the space of smooth sections for a G-equivariant vector bundle over X and that for a G'-equivariant vector bundle over Y a differential symmetry breaking operator (differential SBO for short) if D is G'-intertwining. The F-method is a powerful machinery on the classification and construction of differential SBOs. It allows one to classify and construct differential SBOs by solving a certain system of partial differential equations. The key idea is the Fourier transform of the generalized Verma modules. In this talk we shall overview the F-method in an elementary manner.

Branching Laws and a Generalized Penrose Transform (online)

• Hideko Sekiguchi (University of Tokyo)

Room: Amphitheater Darboux I plan to discuss a generalized Radon-Penrose transform and provide examples of explicit branching laws from the perspective of differential equations.

Introduction to Representation Theory, 3/3

• Birgit Speh (Cornell University)

Room: Amphitheater Darboux The 3 lectures are introduction to representation theory of reductive real Lie groups at the level of the book ”Representation theory of semisimple Lie groups: An overview based on examples” by A.W. Knapp. All the concepts and theorems will be illustrated by examples, but I will give references for the detailed proofs. In each lecture I will give some suggested homework and reading. Lecture 3. We will discuss principal series representations, intertwining operators and the Langlands classification of irreducible representations.

Geometric and Analytic Aspects of Branching Laws, 1/2

• Bent Ørsted (Aarhus University, Denmark)

Room: Amphitheater Darboux In these lectures will be explained some of the basic mathematical theory of group representations, mostly of reductive Lie groups, and their structure via the theory of branching problems and intertwining operators.

Geometric and Analytic Aspects of Branching Laws, 2/2

• Bent Ørsted (Aarhus University, Denmark)

Room: Amphitheater Darboux In these lectures will be explained some of the basic mathematical theory of group representations, mostly of reductive Lie groups, and their structure via the theory of branching problems and intertwining operators.

Branching Problems in Representation Theory, 3/3

• Toshiyuki Kobayashi (The University of Tokyo)

Room: Amphitheater Darboux The concept of symmetries naturally arises in various areas of mathematics and science, including geometry, number theory, differential equations, and quantum mechanics. The more symmetries an object possesses, the better we can understand it through group-theoretic approaches. Branching problems investigate how large symmetries break down into smaller ones, such as fusion rules, using mathematical formulations based on the language of representations and their restrictions. These problems have been studied for over 80 years. In recent years, there has been a surge of research focused on the restriction of continuous symmetries in infinite-dimensional cases, leading to the development of new geometric and analytic methods. In my three lectures, I plan to provide an introduction to the branching problems of infinite-dimensional representations of real reductive groups, such as GL(n, R), using plenty of elementary examples to make the basic concepts and key ideas more accessible. A major goal of my lectures is to capitalize on this momentum and bring together young students and postdocs from diverse mathematical disciplines, encouraging them to develop an interest in this promising field. If time permits, I will also present some open questions in the area.

Workshop: Intertwining operators and geometry Room: Amphithéâtre Hermite See: https://indico.math.cnrs.fr/event/10856/timetable

Workshop: Intertwining operators and geometry Room: Amphithéâtre Hermite See: https://indico.math.cnrs.fr/event/10856/timetable

Workshop: Intertwining operators and geometry Room: Amphithéâtre Hermite See: https://indico.math.cnrs.fr/event/10856/timetable

Workshop: Intertwining operators and geometry Room: Amphithéâtre Hermite See: https://indico.math.cnrs.fr/event/10856/timetable

Workshop: Intertwining operators and geometry Room: Amphithéâtre Hermite See: https://indico.math.cnrs.fr/event/10856/timetable

The Baum-Connes conjecture for real reductive groups 1/10

• Nigel Higson (Penn State University)

Room: Amphitheater Darboux K-theory for C*-algebras A second review of K-theory for C*-algebras: Bott periodicity, stability and long-exact sequences. Really basic C*-algebras: characterization and K-theory. Continuous fields of C*-algebras, and the K-theory groups of a continuous field. Mapping cone continuous fields. Open subsets of the spectrum of a C*-algebra and ideals. Examples related to SL(2,R).

The Baum-Connes conjecture for real reductive groups 2/10

• Nigel Higson (Penn State University)

Room: Amphitheater Darboux The Cartan motion group and the Mackey deformation family The Cartan motion group. The unitary dual and the C*-algebra of the motion group. Structure in the unitary dual from the restricted root system. A filtration of the group C*-algebra by ideals. The deformation construction associated to a Lie group and a maximal compact subgroup. The associated continuous field of group C*-algebras. The Connes-Kasparov morphism in C*-algebra K-theory. Relation to the Baum-Connes assembly map. The example of SL(2,R).

Introduction to the Baum-Connes conjecture 1/10

• Alain Valette (Université de Neuchâtel)

Room: Amphitheater Darboux The Baum-Connes conjecture in a nutshell A corollary: the Kaplansky-Kadison conjecture on idempotents

Introduction to the Baum-Connes conjecture 2/10

• Alain Valette (Université de Neuchâtel)

Room: Amphitheater Darboux K-theory for C*-algebras, again K_0, K_1, stability, Bott periodicity, 6-terms exact sequence

Cuntz’ K-theoretic amenability revisited

• Pierre Julg (Université d'Orléans)

Room: Amphitheater Darboux In the early 1980’s, the question was raised of comparing the K-theory groups of the full and reduced C*-algebras of a group $G$. J.Cuntz has described a condition (K-amenability) implying that they are isomorphic. Note that K-amenability is incompatible with Kazhdan’s property T, and is implied by the Haagerup property, a strong negation of property T. In this talk we shall explain that Cuntz’ condition relies on the construction of a $G$-Fredhom module. We shall give such a module in the example of $\mathrm{SL}_2$ on the fields of $p$-adics (Julg-Valette), of complex numbers (Kasparov) and of real numbers (Fox-Haskel and Julg-Kasparov).

Introduction to the Baum-Connes conjecture 3/10

• Alain Valette (Université de Neuchâtel)

Room: Amphitheater Darboux Kasparov theory Hilbert C*-modules, operators, compact operators; KK-cycles

Introduction to the Baum-Connes conjecture 4/10

• Alain Valette (Université de Neuchâtel)

Room: Amphitheater Darboux Equivariant Kasparov theory KK_G-cycles, the ring R(G), the descent map

Introduction to the Baum-Connes conjecture 5/10

• Alain Valette (Université de Neuchâtel)

Room: Amphitheater Darboux Proper actions and the classifying space for proper actions

Introduction to the Baum-Connes conjecture 6/10

• Alain Valette (Université de Neuchâtel)

Room: Amphitheater Darboux The assembly map and the BC conjecture (with and without coefficients)

The Baum-Connes conjecture for real reductive groups 3/10

• Nigel Higson (Penn State University)

Room: Amphitheater Darboux The imaginary part of the infinitesimal character Definition of the infinitesimal character. The infinitesimal character of a parabolically induced representation. Continuity of the infinitesimal character. The imaginary part of the infinitesimal character. A filtration of the reduced C*-algebra of a real reductive group by ideals. Some low-real-rank examples. A theorem of Bruhat and Harish-Chandra about irreducible parabolic induction. Characterization of the fibers of the imaginary part of the infinitesimal character using tempiric representations. Discrete series representations versus tempiric representations. Characterization of the subquotients for the filtration of the reduced group C*-algebra by ideals. The Mackey bijection.

The Baum-Connes conjecture for real reductive groups 4/10

• Nigel Higson (Penn State University)

Room: Amphitheater Darboux More on the discrete series Harish-Chandra’s Schwartz space. Tempered representations. The Harish-Chandra-Langlands principle. Asymptotic expansions of matrix coefficients and idempotents in the reduced group C*-algebra. Speculations about a C*-algebraic approach to the discrete series and the Harish-Chandra principle.

The Baum-Connes conjecture for real reductive groups 5/10

• Nigel Higson (Penn State University)

Room: Amphitheater Darboux Proof of the Bruhat-Harish-Chandra theorem Distributions from intertwiners between parabolically induced representations. Bruhat decomposition. Bounding the dimensions of the spaces of equivariant distributions. Proof of the theorem.

The Baum-Connes conjecture for real reductive groups 6/10

• Nigel Higson (Penn State University)

Room: Amphitheater Darboux The Connes-Kasparov isomorphism from the perspective of representation theory The Mackey deformation family and continuity of the rescaled imaginary part of the infinitesimal character. A filtration of the deformation continuous field by ideals. Rescaling automorphisms, embedding morphisms, and computation of the subquotient continuous fields. Reformulation of the Connes-Kasparov isomorphism. Vogan’s theorem.

Introduction to the Baum-Connes conjecture 7/10

• Alain Valette (Université de Neuchâtel)

Room: Amphitheater Darboux Status of the BC conjecture

Introduction to the Baum-Connes conjecture 8/10

• Alain Valette (Université de Neuchâtel)

Room: Amphitheater Darboux BC vs CK

Introduction to the Baum-Connes conjecture 9/10

• Alain Valette (Université de Neuchâtel)

Room: Amphitheater Darboux The "Dirac-dual Dirac" method and the gamma-element

Introduction to the Baum-Connes conjecture 10/10

• Alain Valette (Université de Neuchâtel)

Room: Amphitheater Darboux Nishikawa's recent approach

The Baum-Connes conjecture for real reductive groups 7/10

• Nigel Higson (Penn State University)

Room: Amphitheater Darboux Kasparov’s gamma element for SL(2,R) Practical person’s guide to the Baum-Connes conjecture from the point of view of Kasparov theory: Kasparov’s representation ring (as an abelian group), the gamma element, Kasparov’s theorem for connected Lie groups, reduction to $\gamma=1$. First examples where $\gamma$ is equal to 1. Examples when $\gamma$ is not equal to 1. Julg’s proposal regarding uniformly bounded representations. Kasparov’s homotopy for SL(2,R).

The Baum-Connes conjecture for real reductive groups 8/10

• Nigel Higson (Penn State University)

Room: Amphitheater Darboux Lafforgue’s Banach version of KK-theory The C*-algebra of a group, versus other convolution Banach algebras. Homotopies of unitary representations versus homotopies of Banach representations. Examples and action of the reduced C*-algebra on $L^p$ spaces. Review of the role of compact operators in Kasparov theory. Dual pairs of Banach spaces and operators of compact type. Lafforgue’s version of Kasparov’s representation ring.

The Baum-Connes conjecture for real reductive groups 9/10

• Nigel Higson (Penn State University)

Room: Amphitheater Darboux Banach modifications of the gamma element The problem with Banach convolution algebras from the point of view of representation theory. The problem with tensor products and unconditional norms. Unconditional norms. Karoubi density. Reduction of the Baum-Connes conjecture to $\gamma=1$. A further technical reduction. The rapid decay property for the Harish-Chandra Schwartz algebra.

The Baum-Connes conjecture for real reductive groups 10/10

• Nigel Higson (Penn State University)

Room: Amphitheater Darboux Rapid decay property and Lafforgue’s proof of the Baum-Connes conjecture The gamma element from Witten’s perturbation of the de Rham complex. Transfer to weighted $L^2$ spaces. A homotopy from the gamma element to 1, almost. Putting everything together.

Structure of tempered homogeneous spaces 1/4

• Toshiyuki Kobayashi (The University of Tokyo)

Room: Amphitheater Darboux

Structure of tempered homogeneous spaces 2/4

• Toshiyuki Kobayashi (The University of Tokyo)

Room: Amphitheater Darboux

Structure of tempered homogeneous spaces 3/4

• Toshiyuki Kobayashi (The University of Tokyo)

Room: Amphitheater Darboux

Structure of tempered homogeneous spaces 4/4

• Toshiyuki Kobayashi (The University of Tokyo)

Room: Amphitheater Darboux